Sr Examen
Integral de 1/(x^2+y^2+(z-2)^2)^1/2 dx
Solución
Ha introducido
[src]
1 / | | 1 | ----------------------- dx | ____________________ | / 2 2 2 | \/ x + y + (z - 2) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sqrt{\left(x^{2} + y^{2}\right) + \left(z - 2\right)^{2}}}\, dx$$
Integral(1/(sqrt(x^2 + y^2 + (z - 2)^2)), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___________________\ \
|| | / 1 | 2 2 |
/ ||asinh|x* / ----------------- | for 4 + y + z - 4*z > 0|
| || | / 2 2 | |
| 1 || \ \/ 4 + y + z - 4*z / |
| ----------------------- dx = C + |< |
| ____________________ || / ____________________\ |
| / 2 2 2 || | / 1 | 2 2 |
| \/ x + y + (z - 2) ||acosh|x* / ------------------ | for 4 + y + z - 4*z < 0|
| || | / 2 2 | |
/ \\ \ \/ -4 - y - z + 4*z / /
$$\int \frac{1}{\sqrt{\left(x^{2} + y^{2}\right) + \left(z - 2\right)^{2}}}\, dx = C + \begin{cases} \operatorname{asinh}{\left(x \sqrt{\frac{1}{y^{2} + z^{2} - 4 z + 4}} \right)} & \text{for}\: y^{2} + z^{2} - 4 z + 4 > 0 \\\operatorname{acosh}{\left(x \sqrt{\frac{1}{- y^{2} - z^{2} + 4 z - 4}} \right)} & \text{for}\: y^{2} + z^{2} - 4 z + 4 < 0 \end{cases}$$
Respuesta
[src]
/ 1 \
asinh|-----------------------------------------|
| ______________________________________|
| / / 2 2 \ |
\\/ polar_lift\y + polar_lift (-2 + z)/ /
$$\operatorname{asinh}{\left(\frac{1}{\sqrt{\operatorname{polar\_lift}{\left(y^{2} + \operatorname{polar\_lift}^{2}{\left(z - 2 \right)} \right)}}} \right)}$$
=
=
/ 1 \
asinh|-----------------------------------------|
| ______________________________________|
| / / 2 2 \ |
\\/ polar_lift\y + polar_lift (-2 + z)/ /
$$\operatorname{asinh}{\left(\frac{1}{\sqrt{\operatorname{polar\_lift}{\left(y^{2} + \operatorname{polar\_lift}^{2}{\left(z - 2 \right)} \right)}}} \right)}$$
asinh(1/sqrt(polar_lift(y^2 + polar_lift(-2 + z)^2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.